Work
Created by John Bennewitz This article was created to educate a reader on the fundamental topic of work, by including various definitions of work (both thermodynamic & mechanical) and discussing the differences between reversible/irreversible/impossible processes. Also. certain engineering examples of work drawn from James Prescott Joule's experiments are also considered to provide further understanding of the topic. __TOC__ Definition Thermodynamically, work is defined as a process in which energy is transferred across a system boundary. Work can be thought of as an effect of one system imposed onto another (e.g. a thermodynamic system acting on its surroundings)[1]. The presence of work performed by a system can be realized from the following statement, "Work is done by a system on its surroundings if the sole effect of everything external to the system could have been produced by the raising of a weight" [2]. It should be noted that from this statement, it is not necessary for a system to physically raise a weight (or essentially have a force act across a distance) to have work present, but rather that the same effect on the system could have ''been replicated by raising a weight. Mechanically, '''work '''is defined as an energy transfer process in which a force acts across a distance. This type of work is dictated by the following simple mathematical expression: W = \int\bold{F} \cdot \mathrm{d}\bold{s} , where F is the applied force & dS is the displacement of the acting force. From this mathematical expression, it is apparent that if both the displacement and force act in the same direction, the work produced is positive; negative work on the other hand is created when the force and displacement act 180° opposite of each other. It should also be noted that in a cyclic process where a force is acted across a distance and then returned to its original position (causing dS = 0), there is ''no mechanical work performed. The units of work are generally either a Joule (when working in S.I. units) or foot-pound (when working in English units). Some physical significance can be drawn from the units of work. While considering the foot-pound, it is readily apparent that work involves a force acting over a distance (foot-pound = distance*Force). For the S.I. standard unit, a bit more analysis needs to be performed to reach the same understanding. Essentially, a Joule is a Newton-meter (J = N*m). In the S.I. system, the unit of force is a Newton, so the same form can then be reached: J = N*m = Force*distance). Joule's Experiments To further understand this process of energy transfer, it is worth examining a series of experiments performed by James Prescott Joule in the 1840's. In his experiments, he aimed to determine the mechanical energy (i.e. mechanical work) necessary to raise the temperature inside a water bath by 1 °F [3]. For these experiments, there existed an insulated bath of water, which would be configured in a way to be effected by various sources of work. These experiments demonstrate a fundamental understanding of certain variations of work and how they are able to alter the state of the system through the First Law of Thermodynamics. The First Law of Thermodynamics states that energy is conserved throughout a system, and that when one form of energy (work, heat or internal energy) is altered, the others change correspondingly to permit dE = 0. Upon studying the First Law of Thermodynamics, it is important to understand that not only is energy conserved throughout a system, but that the various types of energy transfer (work, heat or internal energy) are independent of one another and have different characteristics. From this, it can be said that the ability for work to be an organized energy transfer process compared to heat transfer is an extremely important characteristic of work which is demonstrated by Joule's experiments. His tests are explained below in further detail: First Experiment The first set-up consisted of a propeller and weight apparatus to supply work to the water bath. Essentially, the work necessary to raise the temperature of the water bath for this experiment was able to be quantified by measuring the vertical displacement (dy) of the mass (of a known value). By applying a known mass value to the pulley, only the vertical displacement needed to be measured (simply using a length measuring device) in order to quantify the amount of work. It was determined that 773 ft*lbf was required to raise the temperature in the water bath by 1 °F [3].. Second Experiment The second experiment consisted of two masses, one directly on top of another. Also, an outward force was applied to the top mass (creating heating of the water bath due to friction). Overall, the work necessary to raise the temperature of the water bath for this experiment was quantified by measuring the horizontal applied force (of a known value) across the displacement (dx). By applying a known force, again, only the displacement needed to be measured in order to quantify the amount of work. It was determined that 775 ft*lbf was required to raise the temperature in the water bath by 1 °F [3].. Third Experiment The third experiment consisted of a piston cylinder sytem located inside the water bath. From this set-up, a force (of a known value) was applied to the piston to compress the cylinder (creating heating of the water bath due to simple compressive work). Essentially, the work necessary to raise the temperature of the water bath for this experiment was determined by quantifying the horizontal displacement (dx) of the force applied to the cylinder. It was determined that 793 ft*lbf was required to raise the temperature in the water bath by 1 °F [3].. From these experiments, it is apparent that by applying various types work to a system, it is possible to generate heat within the system. Essentially, due to all three of the systems being insulated, there is no heat transferred from the systems to the surroundings. Then, from the First Law of Thermodynamics, when work is applied to the system, the internal energy of the water bath must increase. Also, it should be noted that the work required to raise the bath temperature 1 °F for each of the various external work sources is essentially constant across the three experiments. Due to the constant volume nature of the systems in Joule's Experiments, the amount of heat generated in the system which raises the water bath 1 °F is now realized as the specific heat at constant volume for water (which Joule measured to be approximately 781 (ft*lbf)/(lbm*R) and is now accepted to be 778 (ft*lbf)/(lbm*R)) [3]. Simple Compressible Substance When considering thermodynamics, it is often useful to apply the simple compressible substance assumption to a flow system. Under this assumption, the only type of work that can naturally occur in the flow system is fluid compression work. "This term designates substances whose surface effects, magnetic effects, and electrical effects are insignificant when dealing with compressible substances" [1]. Fluid compression work, or "pdV" work as it is sometimes called, is a type of reversible work which is quantified by the following equation: W = pdV , where p is the pressure of the fluid & dV is a finite change in volume of the fluid. From this relation, it is apparent that when a fluid is expanded (positive dV value), work is released from the system, and while the fluid is compressed (negative dV value), work must be supplied to the system. Thus, this relation will adequately describe the compression and expansion of a simple compressible fluid. Once applying this assumption to the First Law of Thermodynamics, many thermodynamic systems (some of which are applicable to the field of aerospace engineering) can be simplified and solved. An example of one of these processes would be the Brayton Cycle, which is a thermodynamic cycle used in the analysis of turbine engines. Reversible / Irreversible / Impossible Processes Upon discussing the concept of work in thermodynamics, it is important to consider the types of processes a system can undergo. Thermodynamic processes can be divided into the three following designations: Reversible, Irreversible and Impossible. Differentiation between these three types is directly attributed to the Second Law of Thermodynamics and the production of entropy. Reversible Process Essentially, a reversible process is an ideal thermodynamic process. In other words, when a process is reversible, no change on a system or its surroundings has taken place [1]. Thus, the process that initially took place is able to be completely reversed with no net change in the entropy of the system (i,e. Entropy Production, Ps = 0). Due to the entropy staying constant, reversible processes are the most efficient as opposed to processes with entropy generation. One example of a reversible process is an extremely slow process (e.g. upon considering combustion, an example would be an ideal gas mixture being slightly altered and having enough time to reach equilibrium) [3]. Reversible Process: \Delta s = Ps = 0 Irreversible Process An irreversible process is one that cannot be reversed and thus permanently changes the system and/or its surroundings [1]. Due to the Second Law of Thermodynamics, for an irreversible process, entropy is produced and the overall efficiency of the process is decreased. As opposed to a reversible process, there is a net change in the entropy of the system. Almost all thermodynamic processes are irreversible, especially those of interest to aerospace propulsion engineers. One example of an irreversible process would be the mixing of two gas species (a fuel and oxidizer) during combustion. Irreversible Process: Ps \ge 0 Impossible Process As the name suggests, an impossible process is one that cannot occur due to violations of the principles of thermodynamics. Essentially, the definition of an impossible process can be viewed as a direct extension of the Second Law of Thermodynamics. From the Second Law, it can be said that entropy of a system can only stay constant or be created during a thermodynamic process; thus, entropy can never be decreased across a process. For an impossible process, entropy generation is said to be negative (leading to an overall decrease in the entropy of the system). Although there are no actual impossible processes that exist, a theoretical example from propulsion would be a weak detonation. For a weak detonation, it would be necessary to accelerate a flow from subsonic conditions to supersonic through only heat addition (the violation is a result of heat addition, always driving flow conditions to M=1) [4]. Impossible Process: Ps < 0 References 1 Sonntag, Richard E., Claus Borgnakke and Gordon J. Van Wylen. Fundamentals of Thermodynamics: Sixth Edition. John Wiley & Sons, Inc., 2003. 2 Bejan, Adrian, George Tsatsaronis and Michael Moran. Thermal Design & Optimization. John Wiley & Sons, Inc., 1996. 3 Denbigh, Kenneth. The Principles of Chemical Equilibrium: Fourth Edition. Cambridge University Press, 1997. 4 Turns, Stephen. Introduction to Combustion: Concepts and Applications. New York: McGraw Hill.